Question

In a recent​ survey, 145 of 457employees responded that they typically took work with them on vacation and 68 responded that there are unwritten and unspoken expectations that they stay connected. Complete parts​ (a) through​ (d) below.

a. Construct a 95​%confidence interval estimate for the population proportion of employees who typically take work with them on vacation.

​(Round to four decimal places as​ needed.)

b. Construct a 95​ confidence interval estimate for the population proportion of employees who said that there are unwritten and unspoken expectations that they stay connected.

​(Round to four decimal places as​ needed.)

c. Interpret the intervals in​ (a) and​ (b). Which of the following is the correct interpretation for the interval in​ (a)?

A.You can be 95​%confident that the population proportion of all employees that say there are unwritten and unspoken expectations that they stay connected on vacation lies within the interval in​ (a).

B.You can be 95​% confident that the population proportion of all employees that take work with them on vacation lies within the interval in​ (a).

C.You can be 95​% confident that the population proportion of the employees surveyed that say there are unwritten and unspoken expectations that they stay connected on vacation lies within the interval in​ (a).

D.You can be 95​% confident that the population proportion of the employees surveyed that take work with them on vacation lies within the interval in​ (a).

Which of the following is the correct interpretation for the interval in​ (b)?

A.You can be 95​% confident that the population proportion of the employees surveyed that say there are unwritten and unspoken expectations that they stay connected on vacation lies within the interval in​ (b).

B.You can be95​% confident that the population proportion of the employees surveyed that take work with them on vacation lies within the interval in​ (b).

C.You can be 95​% confident that the population proportion of all employees that take work with them on vacation lies within the interval in​ (b).

D.You can be 95​% confident that the population proportion of all employees that say there are unwritten and unspoken expectations that they stay connected on vacation lies within the interval in​ (b).

d. Explain the difference in the results in​ (a) and​ (b). Choose the correct answer below.

A.The two confidence interval estimates do not overlap because the two separate point estimates are close enough together.

B.The two confidence interval estimates overlap because the two separate point estimates are close enough together.

C.The two confidence interval estimates do not overlap because the two separate point estimates are far enough apart.

D.The two confidence interval estimates overlap because the two separate point estimates are far enough apart

Answer 1

(a)

n = 457    

p = 145/457 = 0.317286652    

% = 95    

Standard Error, SE = √{p(1 - p)/n} =    √(0.317286652078775(1 - 0.317286652078775))/457 = 0.021771432

z- score = 1.959963985    

Width of the confidence interval = z * SE =     1.95996398454005 * 0.0217714321323338 = 0.04267122

Lower Limit of the confidence interval = P - width =     0.317286652078775 - 0.0426712228712323 = 0.27461543

Upper Limit of the confidence interval = P + width =     0.317286652078775 + 0.0426712228712323 = 0.35995787

The confidence interval is [0.2746, 0.3600]

(b)

n = 457    

p = 68/457 = 0.148796499    

% = 95    

Standard Error, SE = √{p(1 - p)/n} =    √(0.148796498905908(1 - 0.148796498905908))/457 = 0.016647727

z- score = 1.959963985    

Width of the confidence interval = z * SE =     1.95996398454005 * 0.0166477274402891 = 0.03262895

Lower Limit of the confidence interval = P - width =     0.148796498905908 - 0.0326289462074058 = 0.11616755

Upper Limit of the confidence interval = P + width =     0.148796498905908 + 0.0326289462074058 = 0.18142545

The confidence interval is [0.1162, 0.1814]

(c)

B.You can be 95​% confident that the population proportion of all employees that take work with them on vacation lies within the interval in​ (a)

D.You can be 95​% confident that the population proportion of all employees that say there are unwritten and unspoken expectations that they stay connected on vacation lies within the interval in​ (b)

(d) C.The two confidence interval estimates do not overlap because the two separate point estimates are far enough apart

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